IME march 2007 workshop nebraska

The Mathematics Semester for future elementary school teachers... The Mathematics SemesterMath Matters, an NSF-funded CCLI grant began in 2000, and became The Mathematics Semester in Fal

The Mathematical Education of K-8 Teachers

at the University of Nebraska-Lincoln,

a Mathematics – Mathematics Education

Partnership

Jim Lewis and Cheryl Olsen University of Nebraska-Lincoln

The Mathematics Semester for future elementary school teachers

The Mathematics Semester

Math Matters, an NSF-funded CCLI grant began in 2000, and became The Mathematics Semester in Fall 2003

Vision

• Create a mathematician – mathematics educator

partnership with the goal of improving the mathematics education of future elementary school teachers

• Link field experiences, pedagogy and mathematics

instruction

• Create math classes that are both accessible and useful for future elementary school teachers

The Mathematics Semester

(For all Elementary Education majors starting Fall 2003)

MATH

• Math 300 – Number and Number Sense (3 cr)

PEDAGOGY

• TEAC 308 – Math Methods (3 cr)

• TEAC 351 – The Learner Centered Classroom (2 cr)

FIELD EXPERIENCE

• TEAC 297b – Professional Practicum Exper (2 cr)

(a field experience in a local elementary school)

– Students are in an elementary school on Mondays and

Wednesdays (four hours/day)

– Math 300 & TEAC 308 are taught as a 3-hour block on Tuesday and Thursday

– TEAC 351 is taught on site at a participating elementary school

A look inside The Mathematics Semester

• Curriculum materials

• Homework to develop mathematical habits of mind

• Professional writings

• The curriculum project

• Learning and Teaching Project

• Activities at the elementary school

– Teaching a Math Lesson

– Child Study

Curriculum Materials

• Sowder, J et al (2007) Reasoning about Numbers and

Quantities W H Freeman (prepublication copy)

• Schifter, D., Bastable, V., & Russell, S.J (1999)

Number and operations, part I: Building a system of

tens Parsippany, NJ: Dale Seymour.

• Reys, Lindquist, Lambdin, Smith, & Suydam (2007)

Helping Children Learn Mathematics John Wiley &

Sons

• Lampert, M (2001) Teaching problems and the

problems of teaching New Haven, CT: Yale University.

A problem to get started

Making change

What is the fewest number of coins that it will take to make 43 cents if you have available pennies, nickels, dimes, and quarters? After you have solved this

problem, provide an explanation that proves that your answer is correct?

How does the answer (and the justification) change if you only have pennies, dimes, and quarters available?

Note: We first encountered this problem in a conversation with Deborah Ball.

A Typical Weekly Homework All Shook Up

Five couples met one evening at a local restaurant for dinner Alicia and her husband Samuel arrived first As the others came in some shook hands and some did not

No one shook hands with his or her own spouse At the end Alicia noted that each of the other 9 people had

shaken the hands of a different number of people That

is, one shook no one's hand, one shook one, one shook two, etc., all the way to one who shook hands with 8 of the people How many people did Samuel shake hands with?

Note: This problem is a slightly modified version of Exercise #14, page 32, in The Heart of Mathematics, by Burger and Starbird, Key Curriculum Publishing, 2005

Professional Writings

Dear Math Professors,

We are 1st and 2nd graders in Wheeler Central Public School in

Erickson, Nebraska We love to work with big numbers and have been doing it all year! Every time we read something with a big

number in it we try to write it Then our teacher explains how to write

it We are getting pretty good at writing millions and billions!

We have a problem that we need your help with We were reading amazing ‘Super Mom’ facts in a Kid City magazine It told how many eggs some animals could lay We came across a number that we don’t know It had a 2 and then a 1 followed by 105 zeros!! We

wrote the number out and it stretches clear across our classroom!

We know about a googol We looked it up in the dictionary A googol has 100 zeros Then what do you call a number if it has more than

100 zeros? Is there a name for it?

Another problem is that we learned about using commas in large numbers In the magazine article they used no commas when

writing this large number That confused us Also, if you write a

‘googol’ with 100 zeros, how do you put the commas in? It doesn’t divide evenly into groups of 3 zeros There will be one left over.

We appreciate any help you can give us solving this “big” problem Thank you for your time.

Sincerely,

Mrs Thompson’s 1st & 2nd graders

Megan Kansier, Mark Rogers

Marcus Witt, Ashley Johnson

clipping from Kid City magazine

Apple Of My Eye

The tiny female apple aphid is a champ

as an egg-layer This insect can lay as

What do Math Teachers Need to Be?

Read “What do Math Teachers Need to Be?”* by Herb Clemens, a mathematics professor at The Ohio State University Where does your own practice of teaching mathematics stand in relationship to

what Clemens says mathematics teachers need to be: unafraid,

reverent, humble, opportunistic, versatile, and in control of their

math If Clemens came to your classroom and watched you teach

math, how would he answer his question: Can this teacher teach it [math] with conviction, and with some feeling for its essence?

Explain

* Published in 1991 in Teaching academic subjects to diverse

learners (pp 84-96).

M2 Innovations Professional Writings

Curriculum Project

The goal is to investigate a new mathematical area of the

elementary curriculum and consider what teachers need to know as well as what children need to learn the topic in the deep and

meaningful ways suggested by the NCTM Standards (2000).

1) Pick a topic: data analysis and probability, geometry, reasoning and proof, or algebra.

2) Read, analyze, and synthesize the mathematical topic in the NCTM Standards.

3) Analyze and synthesize the topic in a set of reform curriculum

materials (Everyday Math, Trail Blazers, Investigations, local

curriculum).

4) What do teachers need to know to teach this?

5) Create 5 math problems that would help teachers learn Create 5 math problems that would help children learn.

Learning and Teaching Project

Area and Perimeter

The task began with a homework problem It is taken

from Reconceptualizing Mathematics: Courseware for

Elementary and Middle School Teachers, Center for

Research in Mathematics and Science Education, 1998.

Is there a relationship between the area and the

perimeter of a polygonal shape made with congruent square regions? (For fixed area, find the minimum and

maximum perimeter For fixed perimeter, find the

minimum and maximum area.) Squares must be joined

complete-side to complete-side The outside “boundary” should be a polygon In particular, this would not permit

a shape with a “hole” in the middle

A couple of weeks later we told our students:

We want to revisit the “Area and Perimeter” problem This is

to be the basis for a mathematics lesson that you will

videotape yourself teaching to one elementary school student

How can you present this task to the student you will teach? How can you set the stage for the student to understand the problem? How far can the student go in exploring this

problem? Remember that you want your student to discover

as much as possible for himself (or herself) But there may be some critical points where you need to guide the student over

an intellectual “bump” so that he (she) can move on to the

next part of the problem.

Finally, produce a report analyzing the mathematics and your

Activities at the elementary school

Math Lesson You will teach a math lesson that

connects to the curriculum in the classroom in which you are working You should make use of the textbook and other resources from your cooperating teacher and what you are learning and reading in the Mathematics

Semester

Child Study This assignment will give you some

experience watching, listening to, probing, and assessing one child’s understanding of several math problems

focused on a particular area of mathematics You will

write a report of your interview and suggest instruction for the child based on the information you gathered in

the interview session

Math in the Middle Institute Partnership

Principal Investigators Jim Lewis, Department of Mathematics Ruth Heaton, Department of Teaching,

Learning & Teacher Education

Barb Jacobson, Lincoln Public Schools Tom McGowan, Chair, Department of

Teaching, Learning & Teacher Education

(Funding began August 1, 2004)

Invest in high-quality teachers

* To improve K-12 student achievement in

mathematics and to significantly reduce achievement

gaps in the mathematical performance of diverse student populations

M2 Goal

M2 Partnership Vision

• Create and sustain a University, Educational

Service Unit (ESU), Local School District partnership

• Educate and support teams of outstanding middle level (Grades 5 – 8) mathematics teachers who will become intellectual leaders in their schools,

districts, and ESUs.

• Provide evidence-based contributions to research

on learning, teaching, and professional

development

• Place a special focus on rural teachers, schools,

and districts

M2 Partnerships

People and Organizations

• All 14 rural Educational

Service Units plus LPS

• 65 Local School Districts

• 91 Schools

• 130 Teachers (4th cohort

will begin summer 2007)

– 29 teachers earned their

Masters Degree 2006

M2 Major Components

• The M 2 Institute, a multi-year (25-month) institute that

offers participants a coherent program of study to

deepen their mathematical and pedagogical knowledge for teaching and to develop their leadership skills;

• Mathematics learning teams, led by M2 teachers and supported by school administrators and university

faculty, which develop collegiality, help teachers align their teaching with state standards, and assist teachers

in examining their instructional and assessment

practices; and

• A research initiative that will transform the M2 Institute and the M2 mathematics learning teams into laboratories for educational improvement and innovation

Math in the Middle Institute

Partnership

• enhancing mathematical knowledge

• enabling teachers to transfer mathematics they have learned into their classrooms

• leadership development and

• action research

Math in the Middle Institute Design

Summer Fall Spring

Wk1 Wk2&3

Yr 1 M800T Teac800 & M802T M804T M805T

Yr 2 M806T Teac801 & Stat892 Teac888 M807T

Yr 3 M808T Teac889/M809T

and the Masters Exam

- A 25-month, 36-hour graduate program

M2 Summer Institute

• Combination of 1 week and 2 week classes

• Teachers are in class from 8:00 a.m - 5:00 p.m

• 32-35 teachers – 5 instructors in class at one time

• Substantial homework each night

• Substantial End-of-Course problem set

– Purpose – long term retention of knowledge gained.

– Presentation of solutions/celebration of success at start of next class.

M2 Academic Year Courses

• Two-day (8:00 – 5:00) on-campus class session.

• Course completed as an on-line, distance

education course using Blackboard and Breeze.

– Major problem sets

– Professional Writings

– Learning and Teaching Projects

– End-of-Course problem set

– Substantial support available for teachers

M2 Institute Courses

• Eight new mathematics and statistics courses designed

for middle level teachers (Grades 5 – 8) including:

– Mathematics as a Second Language

– Experimentation, Conjecture and Reasoning

– Number Theory and Cryptology for Middle Level Teachers

– Using Mathematics to Understand our World

• Special sections of three pedagogical courses:

– Inquiry into Teaching and Learning

– Curriculum Inquiry

– Teacher as Scholarly Practitioner

• An integrated capstone course:

– Masters Seminar/Integrating the Learning and Teaching of

Mathematics

Math 800T - Mathematics as a

Second Language

• The “text” was written by Kenneth and

Herbert Gross of the Vermont

Mathematics Initiative.

• Ken helped us “kick off” our first weekend.

• Innovations (i.e additions)

– Habits of Mind problems

– Learning and Teaching Project

M2 Innovations

“Habits of Mind” Problems

A person with the habits of mind of a mathematical thinker

can use their knowledge to make conjectures, to reason, and

to solve problems Their use of mathematics is marked by

great flexibility of thinking together with the strong belief that precise definitions are important They use both direct and indirect arguments and make connections between the

problem being considered and their mathematical knowledge When presented with a problem to solve, they will assess the problem, collect appropriate information, find pathways to the answer, and be able to explain that answer clearly to others

While an effective mathematical toolbox certainly includes

algorithms, a person with well developed habits of mind

knows both why algorithms work and under what

circumstances an algorithm will be most effective

M2 Innovations

“Habits of Mind” Problems

Mathematical habits of mind are also marked by

ease of calculation and estimation as well as

persistence in pursuing solutions to problems A

person with well developed habits of mind has a

disposition to analyze situations as well as the efficacy to believe that he or she can make progress toward a solution

self-This definition was built with help from Mark Driscoll’s

book, Fostering Algebraic Thinking: A guide for teachers grades 6-10

The Triangle Game

A “Habits of Mind”

Problem

(Paul Sally, U Chicago) Consider an equilateral triangle with points located at each vertex and at each midpoint

of a side The problem uses the set of numbers {1, 2, 3,

4, 5, 6} Find a way to put one of the numbers on each point so that the sum of the numbers along any side is equal to the sum of the numbers along each of the two other sides (Call this a Side Sum.)

– Is it possible to have two different Side Sums?

– What Side Sums are possible?

– How can you generalize this game?

B

D F

C E

A

Select a challenging problem or topic that you have studied in MSL and use it as the basis for a mathematics lesson that you will

videotape yourself teaching to your students

How can you present this task to the students you teach? How can you set the stage for your students to understand the problem? How far can your students go in exploring this problem? You want your students to discover as much as possible on their own, but there may be a critical point where you need to guide them over an

intellectual “bump.”

Produce a report analyzing the mathematics and your teaching

experience.

M2 Innovations Learning & Teaching Projects

Action Research

Each teacher takes TEAC 888, Teacher as Scholarly

Practitioner, an action research course Each teacher

then conducts their own research project and writes a

report about their findings

– “Action research is research done by teachers for themselves; it

is not imposed on them by someone else” (Mills, 2003, p 5,

italics in original).

– In conducting action research, drawing conclusions isn’t about making generalizations for others but about deciding on a course

of action for one’s own teaching.

– In 2006, 31 teachers had 29 different research projects involving

29 IRB documents

– Each teacher posed 3 research questions, used 3 forms of data collection and used at least 5 from their literature review.